A Networked World (Part 1)

The problem

The idea that’s haunted me, and motivated me, for the past seven years or so came to me while reading a book called The Moment of Complexity: our Emerging Network Culture, by Mark C. Taylor. It was a fascinating book about how our world is becoming increasingly networked—wired up and connected—and that this is leading to a dramatic increase in complexity. I’m not sure if it was stated explicitly there, but I got the idea that with the advent of the World Wide Web in 1991, a new neural network had been born. The lights had been turned on, and planet earth now had a brain.

I wondered how far this idea could be pushed. Is the world alive, is it a single living thing? If it is, in the sense I meant, then its primary job is to survive, and to survive it’ll have to make decisions. So there I was in my living room thinking, “oh my god, we’ve got to steer this thing!”

Taylor pointed out that as complexity increases, it’ll become harder to make sense of what’s going on in the world. That seemed to me like a big problem on the horizon, because in order to make good decisions, we need to have a good grasp on what’s occurring. I became obsessed with the idea of helping my species through this time of unprecedented complexity. I wanted to understand what was needed in order to help humanity make good decisions.

What seemed important as a first step is that we humans need to unify our understanding—to come to agreement—on matters of fact. For example, humanity still doesn’t know whether global warming is happening. Sure almost all credible scientists have agreed that it is happening, but does that steer money into programs that will slow it or mitigate its effects? This isn’t an issue of what course to take to solve a given problem; it’s about whether the problem even exists! It’s like when people were talking about Obama being a Muslim, born in Kenya, etc., and some people were denying it, saying he was born in Hawaii. If that’s true, why did he repeatedly refuse to show his birth certificate?

It is important, as a first step, to improve the extent to which we agree on the most obvious facts. This kind of “sanity check” is a necessary foundation for discussions about what course we should take. If we want to steer the ship, we have to make committed choices, like “we’re turning left now,” and we need to do so as a group. That is, there needs to be some amount of agreement about the way we should steer, so we’re not fighting ourselves.

Luckily there are a many cases of a group that needs to, and is able to, steer itself as a whole. For example as a human, my neural brain works with my cells to steer my body. Similarly, corporations steer themselves based on boards of directors, and based on flows of information, which run bureaucratically and/or informally between different parts of the company. Note that in neither case is there any suggestion that each part—cell, employee, or corporate entity—is “rational”; they’re all just doing their thing. What we do see in these cases is that the group members work together in a context where information and internal agreement is valued and often attained.

It seemed to me that intelligent, group-directed steering is possible. It does occur. But what’s the mechanism by which it happens, and how can we think about it? I figured that the way we steer, i.e., make decisions, is by using information.

I should be clear: whenever I say information, I never mean it “in the sense of Claude Shannon”. As beautiful as Shannon’s notion of information is, he’s not talking about the kind of information I mean. He explicitly said in his seminal paper that information in his sense is not concerned with meaning:

Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual message is one selected from a set of possible messages.

In contrast, I’m interested in the semantic stuff, which flows between humans, and which makes possible decisions about things like climate change. Shannon invented a very useful quantitative measure of meaningless probability distributions.

That’s not the kind of information I’m talking about. When I say “I want to know what information is”, I’m saying I want to formulate the notion of human-usable semantic meaning, in as mathematical a way as possible.

Back to my problem: we need to steer the ship, and to do so we need to use information properly. Unfortunately, I had no idea what information is, nor how it’s used to make decisions (let alone to make good ones), nor how it’s obtained from our interaction with the world. Moreover, I didn’t have a clue how the minute information-handling at the micro-level, e.g., done by cells inside a body or employees inside a corporation, would yield information-handling at the macro (body or corporate) level.

I set out to try to understand what information is and how it can be communicated. What kind of stuff is information? It seems to follow rules: facts can be put together to form new facts, but only in certain ways. I was once explaining this idea to Dan Kan, and he agreed saying, “Yes, information is inherently a combinatorial affair.” What is the combinatorics of information?

Communication is similarly difficult to understand, once you dig into it. For example, my brain somehow enables me to use information and so does yours. But our brains are wired up in personal and ad hoc ways, when you look closely, a bit like a fingerprint or retinal scan. I found it fascinating that two highly personalized semantic networks could interface well enough to effectively collaborate.

There are two issues that I wanted to understand, and by to understand I mean to make mathematical to my own satisfaction. The first is what information is, as structured stuff, and what communication is, as a transfer of structured stuff. The second is how communication at micro-levels can create, or be, understanding at macro-levels, i.e., how a group can steer as a singleton.

Looking back on this endeavor now, I remain concerned. Things are getting increasingly complex, in the sorts of ways predicted by Mark C. Taylor in his book, and we seem to be losing some control: of the NSA, of privacy, of people 3D printing guns or germs, of drones, of big financial institutions, etc.

Can we expect or hope that our species as a whole will make decisions that are healthy, like keeping the temperature down, given the information we have available? Are we in the driver’s seat, or is our ship currently in the process of spiraling out of our control?

Let’s assume that we don’t want to panic but that we do want to participate in helping the human community to make appropriate decisions. A possible first step could be to formalize the notion of “using information well”. If we could do this rigorously, it would go a long way toward helping humanity get onto a healthy course. Further, mathematics is one of humanity’s best inventions. Using this tool to improve our ability to use information properly is a non-partisan approach to addressing the issue. It’s not about fighting, it’s about figuring out what’s happening, and weighing all our options in an informed way.

So, I ask: What kind of mathematics might serve as a formal ground for the notion of meaningful information, including both its successful communication and its role in decision-making?

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39 Responses to A Networked World (Part 1)

David, You foreshadowed the answer — the Semantic Web together with a form of predicate logic. The reality is that you have to have discipline to go down this route, as information and knowledge is no longer free-form but best represented in terms of RDF. For a base ontology, a good starting point for earth sciences is SWEET maintained by JPL. I personally use Prolog as the predicate logic language as it matches well with the subject-predicate-object structure of RDF and it has Semantic Web server support via SWI Prolog.

While I agree I want something someone might call a “semantic web”, I don’t agree that what is currently called the semantic web, i.e., the W3C Tim Berners Lee version, is going to work. Nor do I think any form of predicate logic is going to work. I’ve worked with some of those guys, and though I think there mission is admirable (as you point out, it is similar to my own), I think their approach is too ad hoc to work in the way I’m looking for.

I personally think you need something like category theory, whose purpose is to build bridges between different disciplines, keeping them distinct, yet connecting them. The semantic web tends to put everything into one big jumble.

Ontologies can be understood category-theoretically, but existing languages like OWL are not good.

A wiki can be made into a semantic wiki and it is being done currently. If one looks at Wikipedia pages for geographical locations, on the right hand side one usually sees an infobox containing structured data for that specific location. This is derived from the site DBpedia which publishes structured data from Wikipedia in RDF form. What this does is allow semantic querying of Wikipedia’s data.

Just one example of many on how the semantic web is being incorporated into useful applications. Like artificial intelligence in general, many of these technologies are argued as failures up until the point that they are subsumed and disappear into current technology.

One idea would be for the Azimuth Project Wiki to start to use some of the structured data concepts from the Semantic Web. Then Azimuth coding projects and the Azimuth Wiki can use a common source of data.

I’m on the run, but this coincidence (got a relevant book today [2]) reminds me of a strange absence in the Azimuth Project: The mathematical biologist Robert Rosen [1]. This mostly unsung hero seems the Aristotle + Newton (according to Kant’s judgement) + Gödel that modern (complex) science needs.

According to Rosen, the mathematics you look for is (elementary) category theory. And it seems to polish his ideas/philosophy also higher category theory is called for.

The basis of Rosen’s thought is not “information” but the “modelling relation”. He revisits Aristotelioan causality and arrives at a distinction between machine and organism. All very abstract and mathematical (yet elementary). Rosen characterizes organisms as “closed to efficient causation”. And he draws a simple (still cryptic to me) commutative diagram as the basic “model”. There has only recently been a real breakthrough in giving a concrete computer-simulatable example for this diagram [3].

But I haven’t yet found space/time/brain to get more into the details here.

[1] Robert Rosen (1991), Life Itself: A Comprehensive Inquiry Into the Nature, Origin, and Fabrication of Life.
[2] James A. Coffman, Donald C. Mikulecky (2012), Global Insanity: How Homo Sapiens Lost Touch with Reality While Transforming the World
[3] Athel Cornish-Bowden et al. (2013), Simulating a Model of Metabolic Closure, Biol.Theory 8, 383-390 http://bip.cnrs-mrs.fr/bip10/BiolTheory13.pdf

Hi Martin,
Yes, many people have mentioned Robert Rosen to me, and I even read his book, Life Itself, which you referenced. I think he was right that category theory has the best chance of describing many of the real-world situations that have eluded other disciplines.

Unfortunately, Robert Rosen really didn’t know category theory well, and had some errors at the heart of his arguments. For example, his understanding of “internal-hom’s” was flawed. Some colleagues of mine (Aviv Bergman and some of his students) and I went deeply into the mathematics, and found what I would call fatal errors there.

Still, I think Rosen inspired many people, and his basic ideas could still be correct. We are now at a point where there is enough interest, enough knowledge, and enough pain that a new kind of convergence between biology and mathematics might be taken seriously.

I didn’t know about Rosen’s work. What did he get wrong about internal hom’s? You have to study a certain amount of category theory to bump into those! Most amateurs who try to save the world using category theory don’t get nearly that far.

Hi David, thanks! I strongly suspected I would be carrying owls to Athens here. :-) The internal hom thing was sort of what I found suspicious/puzzling. But I’m not trained and practiced enough to have an easy chance to really grasp all that. So, this fatal technical flaw can’t be remedied with higher category theory? Or maybe a “topos of organism”? (Now I missed the train…) [2nd/3rd try from what seems not a Turing machine. Broken tape?]

John, Rosen was around Mac Lane at U. Chicago in the 50s I believe. He was exposed to a fair bit of category theory, though I think he mainly was “inspired” by it like a jazz musician or artist, rather than doing it like a mathematician. When he tried to do it, he did it wrong.

There are examples in his 1959 paper “A Relational Theory Of Biological Systems II”. There is a clear error on page 116, where he says that for any two sets , the evaluation map is an injection. This is not true for X=2, Y=1. This is not a fatal error, but the stuff on page 117 seems to be basically nonsense. I recall working it out in summer 2014, with the help of (Joaquin Pechuan Jorge) and finding fatal flaws.

Other work by Rosen often makes fairly abstruse category-theoretic claims, which, if you trace back the references, go to this paper. See for example “Some realizations of (M,R)-systems and their interpretation”.

I think Rosen had some really good ideas, and believed that CT would be the right language. But he didn’t care that it actually worked as much as he wanted to convince people that it could work. And he was successful at that. I also don’t get the impression that he was being anything but sincere. Anyway, I think he’s right that we can make the connection, but it’s harder than he thought.

For example, “a person has a mother” would be a morphism from the “person” object, to the “mother” object. I called such a linguistic category an olog, playing on the word blog.

I am sorry I don’t see a conceptional difference between ologs and rdf — but then I don’t understand all the categorical technicalities (and frankly I don’t really want to). rdf itself was inspired by (English) grammar: SPO and the concept of linking. So briefly I understand that people want to teach computers grammars, contexts and the like. I haven’t, though, really understood by your post how exactly this formalism might eventually enhance human understanding, apart may be from the fact that showing others your mind map is surely a way of expressing yourself:

In this setup, the simplicial complex of human knowledge should grow organically. Scientists, business people, and other people might find benefit in ologging their ideas and conceptions, and using them to learn from their peers.

but I’d like to know what useful categorical formulations of them might be.

I followed the IEEE standard upper ontology working group lead by John Sowa (author of the classic ‘Knowledge Representation’) which concluded that no such thing was possible (only intermediate ontologies). When the contributions started discussing Heidegger’s ‘becoming’ I think I got an idea of the limits of any possible standard.

When I was building a Rational Rose model for one of the global electronics corporations I read up on anything UML and concluded that the ontology movement happened because Boing had some spare cash to spend on their engineering drawings library (and ‘Standard English’ became SQL).

Fwiw, one of my favourite quotes goes something like “what differentiates the worst of architects from the best of bees is that men (sic) raise buildings in their imagination before they raise them in reality”.

Do some people still think that you can construct an ontology except via some epistemology? (Sorry if this is OT, but philosophical assumptions need to be made explicit.)

It’s in Mac Lane’s book Categories for the Working Mathematician, and I think he was the first to come out and say it.

One important thing to remember, though, is that category theory is sufficiently abstract that many of its concepts can be seen as special cases of each other. Parodying Clarke’s Third Law: “any sufficiently general concept contains all other concepts as special cases”.

For example: you can think of a left adjoint as a special case of a left Kan extension, but you can also think of a left Kan extension as a special case of a left adjoint.

So I would prefer to focus my admiration on adjoint functors. Why? Because they’re simpler.

Hmm, it seems like an olog is just a category for people who speak English:

A basic olog, defined in Section 2, is a category in which the objects and arrows have been labeled by English-language phrases that indicate their intended meaning.

Anyway, I’d never heard of ‘rdf’, but that paper says:

The idea of representing information in a graph is not new. For example the Resource Descriptive Framework (RDF) is a system for doing just that. The key difference between a category and a graph is…

… that a category is a graph where you can compose the arrows. So if RDF describes things using graphs, ologs gain expressive power by describing them using categories.

The paper also says:

Ologs and RDF / OWL. In [Spi2], the first author explained how a categorical database can be converted into an RDF triple store using the Grothendieck construction. The main difference between a categorical database schema (or an olog) and an RDF schema is that one cannot specify commutativity in an RDF
schema.

In other words, if you can’t compose arrows you can’t say a diagram commutes.

Thus one cannot express things like “the woman parent of a person x is the mother of x.” Without this expressivity, it is hard to enforce much rigor, and thus RDF data tends to be too loose for many applications. OWL schemas, on the other hand, can express many more constraints on classes and properties. We have not yet explored the connection, nor compared the expressive power, of ologs and OWL. However, they are significantly different systems, most obviously in that OWL relies on logic where ologs rely on category theory.

Myself and Marco Perez will put a paper on the arXiv within in the next couple days in which we give a definition of olog and olog functor, that is as formal as possible. It’s a category with a linguistic structure, satisfying certain rules that make the words fit the math.

There are some pretty big differences between RDF and ologs (or categories). One of these is as John mentioned: no notion of composition in RDF. An even bigger one is that all notions of functions with their “domain” and “range” in RDF are actually relations: they do not need to be functions.

If you put enough structure in place (using OWL etc.) you can force RDF triple stores to look like free categories. I do understand how to think of SPARQL graph-pattern queries in terms of lifting problems; these are also equivalent to what database people call embedded dependencies. However, I think CT approaches will be more effective than the current W3C approaches with RDF. I do think the W3C guys are very smart, and very good programmers. I just think they’re working with less-effective mathematics.

Before now I’d never thought about how useful category theory could be to describe networking problems with “multiple layers” – there’s a lot of structure that can describe how the category of ants interacts with the category of anthills.

I’d like to hear more about your intuition about what “information” is or isn’t. For example, if a person lies to another person, is that a transfer of information? What are some characteristics you’d expect your definition of information to adhere to?

Ants and ant-hills would be something I’d love to understand better. I’ve read one book by E.O. Wilson, but I’d like to understand better.

My closest attempt was this paper on what I call “mode dependent networks”. Here you have a bunch of nodes, which I call modules for various reasons, which can be formed into networks, to form “higher-level” modules. The modules are running some kind of dynamical system. For example, at any given time, an ant is in some state; incoming perception updates the state, and the ant accordingly takes some action. This is what I mean by a dynamical system: a set $St$, a function $St\times Inp\to St$ and a function $St\to Outp$.

The connection pattern of the network decides how outputs are fed to inputs. In turn, the combined state of all the modules in a network is what determines the connection pattern of the network.

Using operads, you can consider the whole anthill to be another module in a bigger network.

I’ve been thinking a bit about this type of layering. If you are in some symmetric monodial category and have a morphism that is the result of composing and tensoring a bunch of other morphisms there is a sense in which the structure doesn’t change as you ‘zoom-in’ or ‘zoom-out.’

Would you say that operads are the right machinery for hanndling situations where you want to tweak the types or the structure as you ‘zoom-in/out’?

For example, a situation where the components of your system interface nicely together but have some more complicated internal structure.

I’m not sure I’m thinking along the same lines as you guys, but there’s something like an “operadic completion” that I think is used by people like Fulton and MacPherson to describe things like certain compactifications of configuration spaces. This is meant to connect with this “zooming in”: imagine what looks from a distance like a “sprout” (n incoming edges at a vertex and one outgoing edge), which after a zooming in (or infinitesimal blow-up in the sense of algebraic geometry), resolves to a possibly more complex tree with n leaves and an outgoing root. In essence, the space in which we admit all levels of further resolutions until we can’t go any further (binary trees) naturally carries an operad structure.

Hi Todd, I’m not thinking about anything quite that fancy, though it sounds pretty interesting. I’m just thinking about the operadic nature of string diagrams: you can plug a string diagram into each morphism-box of a string diagram.

For example (as I’ll mention in part 3 of this 3-part Azimuth post), the operad for traced monoidal categories is (roughly) the one underlying the monoidal category 1-Cob of oriented cobordisms between 0-manifolds. If you draw an oriented 0-manifold as a box with negatively-oriented points as incoming wires and positively-oriented points as outgoing wires, you’ll see how it works. The only roughness in my “roughly” has to do with object-sets (1-Cob/{*} really gives you the category of traced monoidal categories whose objects are in bijection with the monoid of natural numbers} and 2-cells between traced monoidal functors.

That’s fine; the “zooming-in” idea, where we can put nodes as black boxes under a microscope to reveal more internal structure (somewhat reminiscent of the cover of the Quantum Field Theory book by Peskin and Schroeder) just put me strongly in mind of this. If you find yourself curious about this later, here’s at least one link: http://www.math.umn.edu/~voronov/8390/lec13.pdf There are also ways of describing the “classical” combinatorial operads (associahedral, permutoassociahedral, …) in these terms.

Concerning the mathematics of meaningful information: is meaningful information roughly synonymous to knowledge? This is the impression that I get from your statements like “figuring out what’s happening”. If it’s substantially different from knowledge, then what is the difference and what is an example in which one can see the difference?

So how could we formalize knowledge? I don’t perfectly know, but I think that knowledge has to do with the conceivable actions that an agent is capable of: you know French if and only if you can speak or write in French; you know baking if and only if you can produce a loaf of bread, given enough flour and an oven.

Alas, this definition may not capture all forms of knowledge. For example, what about knowledge about knowledge? Do I have any special ability if I know that you know French?

These are my Saturday morning thoughts. Do they contain meaningful information? I’m not sure!

As for your first question, is “meaningful information” the same as “knowledge”, I’d have to ask what you mean by knowledge. Some people mean “justified true belief”, but I don’t like the “true” part.

I think we have the same idea though, based on your second paragraph. Meaningful information allows us to ask for and find the french word for “bus station”, and using the result of such a query and the signs on the road, maybe I can get myself to the bus station.

What is it to know that this bag in your hand contains flour, and what flour has to do with making bread? Knowledge, in our sense, is something like a network of interrelated ideas. That is, it is partly the connection between flour and bread that makes flour what it is. What are these connections?

Imagine the amount of information, and the number of different kinds of connections it takes to get you through an airport. If you consider the steps: walking into the right terminal, finding the right ticket counter, giving your flight number, getting a stub which you put in a place you’ll be able to access easily, putting luggage on a belt (tagged with your destination), giving your license along with your boarding pass to the TSA agent who verifies their agreement, looking up to find a sign that says B17 which is what the ticket says, etc. There’s tons of meaningful information that interlocks in many different ways. Can this be understood formally at all?

I would like to say that Shannon information is, for starters, a decategorification of some deeper concept of information, just like the cardinality of a set or the dimension of a vector space. It merely measures the ‘amount’ of stuff: that’s useful, but we also want to be able to talk about the actual stuff. The paper by Tom Leinster, Tobias Fritz and me makes this quite precise.

As for ‘meaningful’ information, this is much harder to define—and more interesting. Someone said that “information is a difference that makes a difference”. This would be a way to try to distinguish between ‘irrelevant’ information like the positions of all the water molecules in the cell, and ‘relevant’ information like the base pairs in the DNA. The former is a difference that doesn’t make (much of a) difference in the behavior of the cell, while the latter is a difference that makes a difference.

To formalize this mathematically, I think we need something like a model of an agent. I think Naftali Tishby’s work on the action-perception loop is a great step in this direction! Everyone should learn that stuff.

I want to start working on this general kind of thing, without necessarily using the same formalization.

Gregory Bateson was the one you’re quoting: “What we mean by information — the elementary unit of information — is a difference which makes a difference, and it is able to make a difference because the neural pathways along which it travels and is continually transformed are themselves provided with energy. The pathways are ready to be triggered. We may even say that the question is already implicit in them.” (from Wikiquote. Another example might be an abacus: you don’t care exactly where the bead is; you care whether it’s on this side or that of the “big gap”.

I too want to work on this general kind of thing, without necessarily using the same formalization. But I’ll take a look at Naftali Tishby’s work.

To add to this interesting discussion, I thought I might mention James Gibsons’ The Ecological Approach to Visual Perception for a view of information which has takers in the Embodied Cognition community (for a brief on embodied cognition see: M. Anderson, Embodied cognition: A field guide, Artificial Intelligence149 (2003)).

In the book, Gibson mostly talks about perceptual information. Roughly speaking, according to Gibson, (optical) information are invariant structures in energy arrays (structured arrangement of light with respect to a point of observation) dependent on the event that caused it. It is something external to the observer (that becomes available as soon as light fills up a volume). Optical information is not something that is processed or transmitted, rather, a nervous system ‘resonates’ to it (as behaviour) when it passes through a given energy array.

I’d also like to mention a very interesting paper: Thelen et al. The dynamics of embodiment: a field theory of infant perseverative reaching, Behavioral and Brain Sciences 24 (2001). The paper provides an interesting explanation of the A-not-B error from the embodied dynamical systems perspective.

(disclaimer: I am not a cognitive scientist but a math grad student with an unhealthy interest in the subject)

I’m definitely interested in what Gibson’s approach “affords”. And resonance is something I’m looking into. In fact, I’m working with a couple people right now to discuss these very issues. Thanks for the Thelen reference.

By the way, all the stuff I talked about in these posts were my hobby, or “unhealthy interest” you might say. And it eventually turned into much more.

Cool post!
I think that if you want to incorporate semantics into information theory, we need to look at evolution theory: Information that is meaningful to humans is ultimately related to increasing our chance of survival. For example, an extremely meaningful message would be “What out, a tiger on your left.”
What prevents us humans from transmitting useless information to each other is that this waste our time, which decreases our survival chances, so that strategies which result in “meaningful” information transfer survive and have become part of our intellectual heritage.
As you remark, the Internet can develop a brain of its own, no single person can control what information roams the Internet. Luckily the Internet is stiil dependent on humans for its survival. We are the ones who click on links. So for the time being, those Internet sites that are useful (or perceived as useful) to humans will be influential. Others will be doomed to stay undownloaded on servers.
To summarize: The Internet will evolve to a state most useful to humans, who in turn define usefulness as maximizing chance of survival.

Gerard, I very much agree that it should be quite useful to look to evolution. We’re talking about “group selection”, which means that by cooperating better than other groups, we can increase our chances of out-competing them.

Language allows us to coordinate, order-together, our actions. Working together to solve problems is pretty powerful. But ants do this without our kind of language. Our kind of language is also very fungible and extendable. We can use it like sorcerers to do all sorts of amazing things.

How can we account for this power? I haven’t heard of robots who can use, let alone invent, languages for their thinking. I don’t think we understand well enough what language is, or what semantics is, to do so.

The only thing I think I disagree with you about is the idea that just because we know that evolution led to language, i.e., by helping us survive, that this provides any clue about what language or semantics is. I’m not sure if you’re saying that it does or not. But I think by talking about “tigers on your left”, you’re already assuming semantics. We need to know what “tiger”, as a semantic notion, is. Would you agree?

For me, the most useful thing to understand in evolution is the jump from prokaryote to eukaryote, and the jump from eukaryote to multicellular organism. What kind of proto-communication led to these jumps?

I think that when talking about “tigers on your left”, we can get by with just assuming pragmatics of display behaviors, without any prior semantics or knowledge of tigers as an ontological type. The speaker is talking about something in the immediate (or proximal) situation, and drawing the attention of the Hearer to something at issue. By looking to the Hearer’s left (or following the gaze of the Speaker), the Hearer’s attention will focus on something big and scary, and will be able to take action. Maybe this is how the Hearer learns for the first time what the sign “tiger” could be referring to. From the pragmatics of display behaviors, specifically vocalization displays and a powerful declarative long-term memory for the forms and significance of signs, hominids evolve full blown linguistic pragmatics.

Within the species capacity of language pragmatics, I think semantics, in the typical sense of the word as used in logic and math, comes out primarily in relation to using utterances in some proximal situation to draw Hearer (or Reader) attention to some distal situation, past present or future. (Language involves some kind of information channel from proximal situations with utterances to distal situations, actual or hypothetical, classified by the semantic content of the utterance.) Semantics solves the more civilized problem of how can a hominid discover or construe the intended content (“what is said”) of an expression, relying only on the bare minimum of the utterance and shared knowledge of lexical items and constructions. It allows sophisticated speakers to construct utterances that could be understood by the entire speech community using the shared lexical/grammatical knowledge, abstracting away from the specifics of a pragmatic situation with particular speakers and factual circumstances. The semantic content is what can be constructed from the utterance and shared prior knowledge of language, independent of pragmatic details of usage. Mathematics works this way as well, with the additional aspiration that well-defined signs and formulas have a determinate content. In contrast, natural languages thrives on having overloaded signs, with rich polysemous meanings. This is not a design defect, it is what Jon Barwise called the efficiency of language.

I am hopeful that category theory (including the Chu spaces studied by Barwise and Seligman, 1997) and type theory can provide a specification language for both pragmatics and semantics. I am not able to envision a typical scientist or mathematician adopting such a precise language in their blogs, but I can imagine it powering analytical engines beneath a more user-friendly natural-language interface. Such an implemented system could help overcome the pitfalls of ambiguous language by fixing the construal of expressions in terms of a clearly specified framework of mostly-deductive lexical and constructional semantics, with a capability for ampliative or abductive inference in the underlying pragmatics. We could have libraries of clear thinking, from which ordinary scientists and lay people, could construct discourse, agree on problems and coordinate solutions. For example, it could help the selective divestment movement convince listed companies and national governments to gradually write off the “value” of the proven coal and oil reserves, a stranded asset in any world that mitigates climate change.

It is an interesting post.
I read a summary of an article on a research of Rodrigo Quian Quiroga, where a single neuron identify a person; so I thought (superficially) that the same process can be applied to each information in the brain (there are single neuron that are activated by a real, or abstract, concept), so that when happen that a person sees some visual features, then a first level words are activates (the shape of the face, the eye color, the hair color, etc), and a combination in a complex proposition generate the activation of the neuron.
I think that the though is a simple chemical reaction, so that it can be that the brain is an evolution of an elementary process of bacteria, and virus, that reasoning with chemical reaction, and the dna like a chemical brain that contain the possible reactions.
A network of perceptrons could contain the sentences of a grammar, without syntax verbs and adjectives, with a neurons that contain verbs, adjectives, nouns and that are activated by the senses.
If the network is great, and near a critical point, so that a little variation of the sense (or a little variations of the inner neurons) a spontaneous dynamic could be activated, and a reasoning on itself could be activated (like a Ising model), something similar to the consciousness, that could be (in this case) a parallel process on a logical network.

This May, a small group of mathematicians is going to a workshop on the categorical foundations of network theory, or Jacob Biamonte. I’m trying to get us mentally prepared for this. We all have different ideas, yet they should fit together somehow.

Tobias Fritz, Eugene Lerman and David Spivak have all written articles here about their work, though I suspect Eugene will have a lot of completely new things to say, too. Now it’s time for me to say what my students and I have doing.

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